"I am sorry!"
죄송합니다 is more formal than 미안합니다. 미안해 is casual between friends. 죄송합니다 > 미안합니다 > 미안해
They often leave out subjects. Adjectives are also not linked to the verb "to be" in Korean.
I need more about it to understand why 미안해(요) is wrong and 죄송합니다 is the right option.
PS: 罪 in here is actually the Hanja equivalent of 죄, which is a common word in Korean for "sin, guilt".
The second character seems to mean 두렵다, since it's name is "두려울 송, literally "the 송-character for being scared".
So I guess together we get something like "being scared of committing a sin".
So this is how you say it in a more formal way? Is it less formal to say "미안해요"?